The Homotopy Theory of Commutative dg Algebras and Representability Theorems for Lie Algebra Cohomology

Abstract

Building on the seminal works of Quillen and Sullivan, Bousfield and Guggenheim developed a "homotopy theory" for commutative differential graded algebras (cdgas) in order to study the rational homotopy theory of topological spaces. This "homotopy theory" is a certain categorical framework, invented by Quillen, that provides a useful model for the non-abelian analogs of the derived categories used in classical homological algebra. In this masters thesis, we use K. Brown's generalization of Quillen's formalism to present a homotopy theory for the category of semi-free, finite-type cdgas over a field of characteristic 0. In this homotopy theory, the "weak homotopy equivalences" are a refinement of those used by Bousfield and Guggenheim. As an application, we show that the category of finite-dimensional Lie algebras over a characteristic 0 field faithfully embeds into our homotopy category of cdgas via the Chevalley-Eilenberg construction. Moreover, we prove that Lie algebra cohomology with coefficients in a trivial module is representable in this homotopy category, in analogy with the classical representability theorem for singular cohomology of CW complexes. Finally, we show that central extensions of Lie algebras are recovered within our homotopical framework as certain principal cofiber sequences.